On Wilks' problem: Exact recursive formulas via Stein's method for the joint moments of disjoint principal minors of Wishart random matrices

Abstract

In a 1934 Annals of Mathematics paper, Samuel S. Wilks posed the problem of computing joint moments of disjoint principal minors of Wishart random matrices, describing the general case as ``extremely complicated.'' These moments arise in classical multivariate statistics, including multivariate regression, analysis of variance, generalized correlation coefficients, and likelihood ratio testing. We solve Wilks' problem for nonnegative integer exponents and any finite number of disjoint principal minors using Stein's method, specifically the Wishart Stein characterization recently developed by Bailly et al. (2026). The Wishart Stein identity yields a degree-lowering recursion that terminates after finitely many steps and evaluates the desired moment exactly. Numerical experiments validate our recursive formulas against direct Monte Carlo simulations from the Wishart distribution.